Question

$\text{For the linear programming problem:}$

$\text{Maximize}z=3x_1+2x_2$

$\text{Subject to:}$

$-2x_1+3x_2\le 9$

$x_1-5x_2\ge -20$

$x_1,x_2\ge 0$

$\text{The above problem has}$

• A. alternative optimum solution
• B. infeasible solution
• C. degenerate solution
• D. unbounded solution

Answer 1

The correct option is D unbounded solution

$\text{Maximize}z=3x_1+2x_2$

$\text{Subjected to :}$

$-2x_1+3x_2\le 9...\left(1\right)$

$x_1-5x_2\ge -20...\left(2\right)$

$\text{Converting constraint (2) in}\le \text{form,}$

$-2x_1+3x_2\le 9...\left(1\right)$

$-x_1+5x_2\le 20...\left(2\right)$

$\text{Again,}\frac{x_1}{\left(\frac{-9}{2}\right)}+\frac{x_2}{3}\le 1...\left(1\right)$

$\frac{x_1}{(-20)}+\frac{x_2}{4}\le 1...\left(2\right)$


Since this is a problem for Max. z and according to the given constant, the maximum value of the objective function lies at infinity. So it simply means that the problem has an unbounded solution.

$\text{Points to Remember:}$

If according to the given condition, the greatest value of the objective function lies at infinity, then it simply means that the common feasible region is not bounded by a limit on the constraint and the solution will be an unbounded solution.